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Example 4: nominal moment for reinforced concrete beam with changing width

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Determine Mn for beam in figure 1.if fy= 60,000 psi and f c'= 4000 psi.

Figure 1








Assuming a>6 in, compressive block for each part with different width will be different as shown in figure 2 (f=0.85*fc'*b)

0.85*fc'*a*b=As*fy
0.85fc'*a1*b1+0.85fc'*a2*b2+0.85fc'*a3*b3=As.fy
0.85*4000*3*6+0.85*4000*3*12+0.85*4000*a3*18=1.56*4*60,000
61,200a3=190,800
a3=3.117

Figure 2


central for concrete compressive force resultant, Top of beam taken as a reference axis as shown in figure 3

 X=(61,200*1.5+122,400*4.5+190,760.4*7.58)/(61,200+122,400+190,760.4)
X=5.579

 Moment is an equal forcein magnitiude and opposit in direction multiplied by the perpendicular distance between force
Mn=As*fy*(d-x)
Mn=6.24*60,000*(30-5.579)=9,143,222.4lb-in=762k-ft

Figure 3



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